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New CHallenges for (adaptive) PDE solvers: the interplay of ANalysis and GEometry

Periodic Reporting for period 3 - CHANGE (New CHallenges for (adaptive) PDE solvers: the interplay of ANalysis and GEometry)

Reporting period: 2019-10-01 to 2021-03-31

Computer-based simulation of PDEs involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions.
In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used as input in Computer-Aided Engineering systems (CAE) where they are handled and processed for the simulation (often also called analysis). The output of the simulation is then visualized and possibly interpreted with an engineering perspective. This workflow is often repeated many times as part of an optimization loop. We quickly realize that new paradigms have to be identified in order to optimize this workflow as a whole. The representation of geometric entities has been studied in the field of geometric modeling, and often the requirements of shape design are different from those of simulation, which is based on numerical methods for PDEs, developed as a branch of mathematics (and mechanics). The use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-)meshing and re-interpolation techniques that are computationally expensive and result in non-exact geometries as well as inaccurate solutions.
By attacking this bottleneck, this project aims at developing innovative mathematical tools for the numer- ical solutions of PDEs, with the goal of considerably improving the efficiency and the robustness of the overall workflow. This goal is interdisciplinary in nature, and calls upon the tight interaction of researchers from the communities of geometry processing and of numerical analysis of PDEs, in order to find a common ground for the representation and use of geometric quantities. The research has to be developed from both sides, with a common objective:
(i) PDE solvers have to be developed / extended in such a way that the class of geometries and tessellation that they can handle, without losing accuracy or robustness, is as wide as possible. We refer to such a class as the class of analysis suitable geometric models or ASGMs.
(ii) geometric modeling techniques are to be developed in order to construct and manipulate ASGMs.
Thanks to the common work on the definition and rules of ASGMs that lies at the very core of CHANGE, the geometry processing becomes PDE-aware, and vice versa.

If successful, the result of CHANGE will reshape the design to analysis pipeline, which is at the very basis of the conception of practically any industrial project. Thus, if on the short term CHANGE’s results consist of mathematical theorems, proofs and academic testing, on the medium-to-long term the results of CHANGE may drastically reduce the time-to-production on any industrial manufactured objected and will contribute to the development of computational twins. Thus, the optimization cycle of industrialized product will be improved allowing for savings of materials, lighting of structures, while safety will be improved and certified by mathematically coherent model descriptions.
Progresses have been made in several aspects both from the geometric side and the analysis side. Suitable geometric models have been identified and the construction of modellers able to produce manipulate, refine, de-refine these models are being developed. On the other hand, analysis can be performed on geometric descriptions that would have been considered “not admissible” just a few years ago.
Two main methodologies are used to develop our approach: isogeometric analysis and polyhedral discretizations. In both cases, we are now able to design complex geometries and manipulate them. Manipulation includes boolean operations such as union and intersection, as well as (local) refinement and de-refinement of the geometric model.
Moreover, on such models, we have designed “measures” of the quality of the geometric model. Solving PDEs on poor geometric models calls for ad-hoc numerical methods, that possibly uses stabilisation techniques to overcome the lack of quality in the geometric model.
These techniques are being written, as of today, for elliptic problems.
All contributions provided until now are beyond the state of the art and the number of our publications and presentations is a measure of the impact of our work.

The grand challenge that remains open in CHANGE is the construction of a truly multi-level geometric description, where model simplification and reduction become part of an adaptive loop. We moved a few steps in this direction, but a real optimal solution to this problem is missing and will be object of intense study in the next year. We expect to find good solution and to open the way to a number of contributions This is a very interesting and unexplored research direction, as, for the time being, model simplification and reduction are done manually and the decisions on simplification are taken regardless their impact on the subsequent simulation.